analytical and geometrical Methods of Investigation 
95 
3 dly. Let 
X * 
+ 
r 
v' a -j- bx-\- cx z Y a -f by -{- cy z 
■ T 
Let cc -f 
•+-*£.+ ± 
C C 
+ 
c c 
0 . 
2 C 
v,y + 
2C 
V 
4 " 
i. 
v' and r 2 
: 0 , 
« 
4 C 
V c 1 d c dv z j^r 7 ’ 
taking the integrals 
f v_ r2r -v 
c—i | V + V' | = «,» 
tr 
e —Y 
/Y ~T j ? fV-fV'. — r 4 £ — (V+V') £aV / r _ r 4 e—xVc 
z? v r 4-v‘ 4-v v (r -\-v = i — * r e 
V 2 2 
= A, and restoring the values of a; and y, 
2£ ^ i v'(«+ 6 J'+0'*)+ ^ v'(a + te + w*) = A'. 
. By the above operation it appears, that certain algebraical ex- 
pressions, as x Vi —/+y \/i—x\ ~± S' a +byj£ C y* & c> may 
be deduced, which answer the equations f — — ■ 4- f — ^ — &c. 
^ V 1 — X z ‘ J Y \ — y z 
But, strictly speaking, such algebraical expressions are not the 
integrals : they are rather expressions deduced from the true 
integral equations, from which other algebraical expressions, 
besides those put down, might be deduced.* 
* For the integration of this sort of differential equations, see Mem. de Turin. VoL 
IV. p. 98. “ Sur PIntegration de quelques Equations differentielles, dont les indetermi- 
nees sont separees, mais dont chaque Membre en particulier n’est point integrable.” 
In this Memoir are given three different methods of integrating ;r (i — x z ) i ~ 
y* (l — y z )~*'> by circular arcs and certain trigonometrical theorems, by impossible 
logarithms, and by partial integrations. Strictly speaking, all these methods are indi- 
rect; and the two first are only different but circuitous modes of expressing the method 
given in Art. X. See likewise Euler, Calc, integral Vol. II. Novi Comm. Petrop. 
Tom, VI. p. 37. Tom. VII. p. I. It is to be observed, that in the present state of 
analytic science, there is no certain and direct method of integrating differential equa- 
