It was therefore determined to return to Crown Point, and early in the spring to complete the original plan. In • the mean time, Wolfe had appeared before Quebec, and although unsupported by the main army under Amherst, and in defiance of numerous and unforeseen difficulties, achieved by means of a mere detachment what had been the chief object of the expedition. On the 8th of September, M. De Vaudreuil capitulated, surrendering Montreal, and all other places within the government of Canada.
General Amherst remained in America until the lat ter part of 1763, when he returned home. During the administration of Mr Pitt, he received every reward that his great services merited. In 1771, he was appointed governor of the island of Guernsey ; in 1776 he was created baron Amherst of Holmsdale, in the county of Kent, and in 1778 appointed commander in chief of the army of England. Upon a change of the ministry, how ever, the command of the army and the lieutenant gene ralship of the ordnance were put into other hands ; but in consequence of the re-establishment of the staff, he was once more appointed commander in chief, in Ja nuary, 1793. It being at length determined to confide the command to the duke of York, a resignation on the part of his lordship was expected : he had the offer of an earl's coronet on this occasion, which he declined, but in July, 1796, accepted the rank of field marshal. He now returned to his seat in Kent, where he died, on the 3d of August, 1797, in the 81st year of his age. From the preceding sketch of lord Amherst, the princi pal traits in his character may be easily distinguished. To employ the words of an officer, who long served un der him, Alexander Hosack, Esq. " To the arts, talents, and military skill of an able soldier, he united all the prudence, sagacity, and management which his impor tant station as commander of a numerous army, in the midst of the most trying occasions, seemed to require." Wynne's British America, vol. ii. Hist. H05AcK and FRANcis.
the 9th section of his Sectiones illiscellanx, published in his Excrcitationes Mathematicie.
In order to investigate the amicable numbers, let 4x be one of the numbers, and 4yz the other; then, since the aliquot parts of 4.r are 1, 2, 4, x, 2x, we have, by the definition of amicable numbers, 1+2+4+.1.+2x=4yz Or 7+3x=4yz 4//z-7 Hence x= 16yz-23 and multiplying by 4.4x= Since the aliquot parts of 4yz are 1, 2, 4, y, 2y, 4y, z, 2z, 4z, 2yz, we have by the definition 4x=1+2+4+y+2y+4y+z+2:+4z+yz+2yz and by substitution and addition 3 which, after deduction, gives 3:+7 16 y= =34-- z-3 =3+ By assuming y=5, a prime number, we shall have z=3+ t another prime, and from a preceding 5-3 equation, we have x5 a third prime 3 number. Hence 4x=284, and 4yz=220, are the ami cable numbers required; for the aliquot parts 1+2+4 +5+10+11+20+22+44+55+110=284, and 1+2= +4+71+142=220.
By putting 8y:, and 16x 16yz, and 128.v 128yz, neces sarily for the amicable numbers, we shall find, by a similar mode of investigation, that 18,416, and 17,296, arc amicable numbers, and also 9,437,056,9,363,584.
From these investigations, we obtain the following rule, which Schoolen received from Descartes. Assume the number 2, or any POWer of 2, such, that if I be sub tracted from thrice the assumed number, from six times that number, or from 18 times its square, the three remain ders shall be prime numbers. illultiply this last prime number by double the number assumed, and the product will be one of the amicable numbers, the sum of whose aliquot ports Will give the other. Thus, for example, assume the number 2, and 5 will be the first prime number, 11 the second, and 71 the third : The first of these being multiplied by 4, the double of the assumed number gives 284 for one of the amicable numbers. The other amicable number, 220, is found by adding the aliquot part of 284. The properties of amicable numbers have been recently investigated by Mr John Gough. Sce Leybourn's Mathematical Repository, No. 7. N. S. (o)