In the fame manner, by finding the values of x in terms of y, &c. we obtain y = m2p3—2m2p+p3—m3 +2p—i, or = p'm2-2mp1-p2+m3 +am—1 m2((p—1)2 —2)+(p+1)2—2 ̧ p'((m-1)-2)+(m+1)1-2 PROBLEM XII. To find three numbers, the product of any two of which, increased by unit, fhall be a Square. By hypothefis, xy+1 = v2, xx+1 = s2, and yz+1 = w2. I. TRANSPOSING the first equation, xyv2-1, and refolving, xxy = (v+1)(v—1), whence y=mv-m, and v+1 2. AGAIN, transposing the second equation, xz = 52—1, and refolving, xXx = (s+1)(s—1); whence, x = ps-p, and 3. MOREOVER, by the third equation, yz = W2—1; whence, y×% = (w+1)(w−1), and y = qw-q, and w+1 THERE is a remarkable cafe in which the above formulæ do not directly apply, the numerators and denominators vanishing at the fame time. It is when m=1, p= 2, and 9. For, 2m3 pq2 —2mp3q1+2m2q by art. 3. y = = I−2+1 = ; where fore the value of y may be expreffed by any affumed number, ». 2=47+4. Thus, 2, 4, 12; for 2X4+19, 2×12+1 = 25, and 4×12+1 = 49. To find a cube which shall be equal to the product of a square by a given number. By hypothefis, x3 ay2, and refolving, xXx2 = a×y2; whence xma, and y2 = mx2; but x2 = (ma)2, confequently, ym3a, and yxy = maxma; and by a fecond affumption, y = pma, and m'apy; but xma; whence y = px, and To find two numbers, the fum of which shall be a given Square, and the fum of their cubes a fquare. By hypothefis, x+y=a2, and x3+y3 = x2. Dividing the fecond equation by the first, we obtain = x2―xy+ y2, or ———y2 = x2—xy, and resolving into factors, (~+y) (~—y) = x(xy); whence, x = m n(~—y), and 2+y= m(x—y). By Cor. I. IF a 2m2+2m-1, two whole numbers may be always found, the fum of which, and that of their cubes, fhall be fquares. For in this cafe, x = (2m2+2m—1)(m2+2m), y = (2m2+2m—1)(m2—1), and z = (2m2+2m—1)2(m2+m+1.) .THUS, if m2, we fhall find x 88, y = 33, and x = 847. But 88+33 = 121 = (11)2, and (88)3+(33)3 = 717409 = (847)2. Cor. 2. IF y be negative, we fhall obtain two numbers, the difference of which, and that of their cubes, fhall be fquares. a3X p2+pq+q2 If a = 2p2+2pq-q', we fhall obtain 2p2+2pq-q2 whole numbers; for x = (2p2+2pq—q2)(p2+2pq), y = (2p2+2pq.—q2)(q2 —p2), and z = (2p2+2pq—q2)2(p2+pq+q2). THESE examples will probably be thought fufficient to explain the application of this method to the folution of indeterminate problems in general, and to fhew that it is not lefs extenfive, and much more uniform, than those that are commonly in ufe. XV. A DISSERTATION on the CLIMATE of RUSSIA. IN a By MATTHEW GUTHRIE, M. D. Physician to the Imperial Corps of Noble Cadets at St Petersburg, F. R. SS. LOND. and EDIN. With two LETTERS from his Excellency M. EPINUS, Counf. of State, Knt of the Order of St. ANNE, &c. &c. &c. [Read by Mr ROBISON, Nov. 2. 1789.] N a paper published in the second volume of the second decade of the Medical Commentaries of Edinburgh, I mentioned a design of endeavouring to trace the influence of a cold climate on the human body and its diseases, which should form a contrast with the many accounts published of late years relative to the effects of hot climates; and I likewife mentioned my having given a detached piece*, fome years ago, as a commencement of the fubject, in the fixty-eighth volume of the Philofophical Transactions of London, which contains matter necessary to illustrate some parts of the following Dissertation. I was induced to this defign, by having met with nothing of the kind in the course of my reading; and by remarking that, whilst warm countries feem to occupy the attention of many of the Faculty, the more northern regions appear to interest but very few of our learned brethren, although it is but natural to conclude that if one extreme of temperature is found to have much influence, the other can scarcely be entirely with out it. IN *THE title of the Differtation mentioned above, is, The Antifeptic Regimen of the Natives of Ruffia. |