pe NS ee ee ae pg Tae te a) 7° Milano ae 
g 1 Equations. Q77 
m— aa i ¢ 
(m—1)x"-2 a mt ~ en~*, or since u=2", we get a= 
a(Z 
mu d r zl lll eal , 
¢ a ,and \dv} =m(m — 1) = 2 oe —— 7 sa ; by substitu- 
iy) 
y q\ de 
d. 
’ d: 
ting these values of ae 
u 
, and t"=—, In (A) we get bya 
A-1\ dy. B? 
aE Se OS Gee 
d?y 
slight reduction <— ay" a+ a > hag “8 
Reduced to the form of (2) by putting ret ee Sees 
63 = 2q-—-2=-~1, or q=3; -" i Se? ae also a? = 1: 
hence the integral of (xk) is found the same way as that of (2) 
given above ; hence we have y expressed by a known function 
of w, then putting for u its value x”, we have y expressed by a 
knew function of z as required. Ronin, if we put u=e"*, we 
dea, 
dy dydu dy dy dx} _ 
d*y 
ee de du de” due ae dus "a 
a Substituting these values and w=e"* in (¢), we have by a 
By 
small reduction mart (M5 ye udu PE “=0, (2); hence if we 
B? 
put 2pq—qt1=1+>, a? =1, OF a 2q -—2=-—lorg=3, we 
Shall have pitt ; then the value of y is found in terms of x, 
asin integrating (2), and by putting for w its value e"*, y be- 
Comes known in terms of z as required. The equations (h) and 
(*) were proposed in the Mathematical Miscellany by Prof. Peirce, 
at p. 399, first volume; we gave an answer to them in No. 8 of that 
Work, which was incorrect in several particulars, which we shall 
hot stop to notice any further than to observe, that uw, the inde- 
“Pendent variable, when integrating with reference to x in La- 
Croix’s integral, and v of our own, was involved in ove of the 
limits of the integrals, so as to be a function of x or v, which ought 
Not to have been so, but the error was not noticed by us in time 
Sufficient for correction before the solution was published. 
