"Sable two bars gemell or on a chief of the second three caltraps of the first¹"
No crest is mentioned, as given by Thoresby². The motto may be: '"Resurgam"- I shall rise again*

* Note 1  that a  motto is not granted with the Achievement of Arms and can be changed at any time. Mottoes may have been medieval battle cries.
* Note 2 The choice of bars-gemel or barrulets (gemelli-twins) is interesting and significant, being derived from the need to strengthen the frames of the early shields.
* Note 3 The presence of the bars and the caltrops on the shield is meant to be  respectively a symbol of the strengthening of the earliest shields and the spiked bosses thereon.The first bull-hide shields were thus reinforced and studded with metal.

It is possible that the Arms or similar were awarded originally to:
1. Sir William de Miggeley of New Hall, Midgley near Wakefield in the 1300's.(Edward III) or his predecessors.
It was during this time that the caltrap made a resurgence with the change in battle tactics from combat by knights on horseback to offence with the longbow and defence with pits and the caltrap.The simplicity of the arms indicates that it could well have been used in the 1200's or 1300's when chivalric knighthood was paramount.
Edward III had  heraldry systematised. The heralds adjudicated on claims to Arms and rival claims were settled by the King with his Constable and Marshal at a Court of Chivalry.
However, at present there is no direct evidence that this was William's Arms, it is quite possible that there were other arms granted to William during this time. He seems to be quite separate to the Midgley families of  the more western parts of the shire, but studies show he may be a descendant of a Norman-French progenitor.

2. A Midgley of Midgley near Halifax in the 1500's. At this time there was a resurgence of the use of heraldic arms with subsequent "Visitations" by the Royal College of Arms, following the closure of the monasteries, e.g. Glover, in his Visitation of Yorkshire in 1594.
Henry VIII admired a full shield and this produced a flurry of "copy-catting' by the landed gentry of the time. However the earlier arms of  titled knights were kept simple, probably because of the real need to be recognised in battle. By the time of Henry VIII, knights fighting on horseback were virtually obsolete.
A later addition of adding an "heraldic Persian tyger"^3,8 crest sitting upright to the Midgley arms may  indicate the service of a decendant of the original grantee, as a man-tyger pursuivant in the early 1500's and possible state service at Calais:
Tyger Pursuviant  [see glossary below] was a title for the supporters of  Lord Hastings, who from 1471 was the Leiutenant-General of Calais.  Following Edward III's victories at Crecy (1364) and Calais (1367), Edward III established  an "English Wool Marketing Board" at Calais, it encouraged ship building and sea-faring in England.  Calais was held for 200 years when in Queen Mary's reign (1558) it was lost to France. However at this time it was a  financial windfall not a loss as Calais was expensive to maintain, however it was a crushing blow to national pride.

The title of Tyger Pursuivant is first recorded (and in fact is the only record) in 1477 when a letter was sent by Bedingfield from Calais to Paxton in Norfolk⁴.

"The heraldic tyger was a particularly popular charge at the beginning of the 1500's. It originated with the bestiary where it was placed correctly between lion and leopard. The bestiary writers stated that the tyger was exceedingly fierce and swift and that the only way to escape it was to throw down mirrors or looking-glasses. The tyger was then distracted by its own image, believing itself to be one of the cubs, curiously the bestiaries do not mention the stripes; but these were Persian tigers which were rather less pronounced in their striping than their Bengal counterparts, so the tiger came into heraldry as a cat-like creature albeit unstriped. Developments at this stage enhanced its ferocity by making its face more wolf-like, adding serrated ears and a horn or tusk to the end of its nose.The heraldic tyger in its fully evolved form is essentially a creation of heraldry, and bears little resemblance to the tiger of the bestiaries. The former has since been introduced into heraldry in its own right"⁴.
The heraldic tyger is described as possessing a mane of tufts as well as a hooked talon at the nose⁸.

Gules  a fess between six garbs Or.

In a bedroom at Ashday Hall, Southowram, among other carved stone shields is one having a fesse between six garbs. This denotes the marriage of Mary, daughter of William Midgley, Gentleman, of Halifax, with Thomas Holdsworth of Ashday. She died on 25th October, 1710. This is unlikely to be a genuine coat of arms as it is probably unrecorded in H.M. College of Arms, London.

There is a second Midgley Coat of Arms in Halifax Parish Church cut on a blue stone within a raised border, painted over, and fixed to the north wall of the Rokeby Chapel. This lies beneath the second window from the west where are found the arms in memory of Mary Midgley, who died 7th November 1704, daughter of William Midgley, M.A., curate of Sowerby, who died  on 10th May, 1706, and according to Watson, late of Headley near Thornton, Bradford. William the Curate is also described as "Mr. William Midgley the minister at Sowerby who died on the 7th May 1706 aged about 30 of the "palsie" [paralysis]⁷
The arms in the Rokeby chapel are crudely carved and appear to be three bars, and in chief three mullets (five pointed stars) with crest-an heraldic tiger, sejant. There are no colours evident. This coat also appears not to have been recognised.

of Holmes Midgley and  J. Jaques respectively to agnates Martha and Rebecca Greenwood¹, viz:Arms- Sable, a chevron ermine between three saltiers couped argent
Crest- a tiger sejant, Or.
As Early as about 1274 Mabell Midgley, daughter of John Midgley of Midgley was married to Ralph Greenwood, he had previously been married to Edith Midleton daughter of Sir William Midleton Kt. of Stockeld.
See also Greenwood Genealogies

"The choice of a caltrap or chevaltrap as an integral feature of approved Midgley achievements would seem to indicate that they had some experience of its use in the fighting so prevalent in northern parts in earlier times. The caltrap was a four-spiked iron ball, one point of which, when placed on the ground was always erect It was anciently used in war to wound the horses' feet and disorganise the cavalry charges, for instance at the battle of Arbela in 331 B.C. Darius the Persian had caltraps placed on the flanks of his troops, to no avail to Alexander's manoeuvers. Then at Bannockburn in 1314 the English knights were bogged down by pits and caltraps. This sort of metal devilthorn which was primarily designed to rip open the horses hooves, could be equally dangerous to the fighting infantrymen. In the course of one counter attack at the seige of Orleans on Friday 6th may 1429, the steel spine of a caltrap pierced the foot of Joan of Arc, Jean la Pucelle and Joan had to be helped off the scene. After the second battle of St. Albans in 1461, during the Wars of the Roses, Andrew Trollope when he was about to be knighted by Queen Margaret, jokingly showed his foot that had stumbled over a caltrap"¹.

Arms- Sable two bars gemel Or and on a chief of the second, three caltraps of the first. 

Compare this with the similarity of crossed keys, wards down [hence the term 'downwards'] on the arms of William Midgley of Rochdale  which may be a more correct interpretation. The Rochdale arms appear to incorporate spinning wheels of cotton.

ii) Midgley (Scholesmoor, Bradford). The crest is a heraldic tiger sejant, holding a caltrap between the paws^{1, 8}.
  iii) Midgley (Rochdale)².The crest is a heraldic tyger sejant resting dexter on a caltrap. 
 

                                                                        .

Another more recent achievement is one patented in South Africa by John Franklin Midgley Viz:
Arms-sable two bars gemel Or on a chief of the first between 6 caltrops of the second  Ar. placed 3, 2, 1.
Crest- Demi (Bengal) tiger proper holding caltrap in each paw.

John F. Midgley in Midgleyana says that he was not in a position to state how many Midgley lines may be armigerous, so it is possible that we may find other grants or variations.

Other references which could prove useful:
West Riding Churches and Monuments, Roger Dodsworth and Visitation of Yorkshire 1665-1666, William Dugdale (giving  heraldic manuscripts from both the 1500's and 1600's) - both held at the Yorkshire Archaeological Society.
# Note: John Watson [1724-1783] Historian and antiquary. Born in Prestbury, Cheshire, he attended Brasenose College, Oxford, and later became curate at Halifax 1750-4, and incumbent at Ripponden 1754-66. In 1759, he was elected a Fellow of the Society of Antiquaries, London. He published several pamphlets and books, including his History and Antiquities of The Parish of Halifax in Yorkshire in 1775, and Memoirs of the Earls of Warren and Surrey in 1776 -Calderdale web pages

Dexter- the right side of a shield as seen by the holder.
Ermine- background pattern, black tails on white.
Fess/fesse-central  horizontal strip about one-third the height of the shield.
Garbs- sheaves of wheat.
Gemels (gemells)- twinned horizontal bars usually borne in pairs.
Gules-red  background colour.
Mullets- five pointed stars. Very early ones are pierced in the centre and are identical to the spurs on the owners effigy on tombs, brass plates etc.
Pursuivant- a state or royal messenger, the third highest heraldic officer (Fr. poursiure-to pursue)
Sable- black colour of an animal coat.
Saltier/saltaire/saltire- crossed diagonally.
Sejant-  animal sitting upright.

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