138 
REPRODUCTION OF EULER’S MEMOIR OF 1758 
[404 
and substituting these values in the differential equation 
dv L cos l M cos m + N cos n 
du p q r 
the equation to be integrated becomes 
^ (L 2 A 2 q 2 r 2 + M 2 B 2 r 2 p 2 + N 2 G 2 p 2 q 2 ) = LMN3) (Ap 2 + Bq 2 + Or 2 ) - LMNv {A 2 p 2 + B 2 q 2 + CV) 
el'll 
4- (L 2 Aq 2 r 2 + M 2 Br 2 p 2 + N 2 Cp 2 q 2 ). 
pqr 
Now substituting for p, q, r their values, we have 
L 2 A 2 q 2 r 2 + M 2 B 2 r 2 p 2 + N 2 C 2 p 2 q 2 = ЕА 2 Ж + М 2 В 2 Ш + №С 2 Ш - 2LMNu (2U 2 + 53Б 2 + SO 2 ), 
L 2 Aq 2 r 2 +M 2 Br 2 p 2 + N 2 Gp 2 q 2 =Ь 2 АШ + М 2 ВШ +№СШ -2LMNu{%A +ЪВ + SO), 
p 2 + q 2 + t 2 = 21 +53 + S + 2 (L + M + N) u, 
Ap 2 + Bq 2 + Or 2 = 2Ы + 53J3 +SO, 
A 2 p 2 + B 2 q 2 + G 2 r 2 = 2L4 2 + 53 B 2 + SO 2 : 
and writing for shortness 
21+ 53+ & = E, 
2IA+ 53Я + £C = F, 
2IA 2 + 53Б 2 + S0 2 =0, 
L 2 A 53S + М 2 ВШ + wem = H, 
1AA 2 53S + M 2 B 2 m + iV 2 0 2 2l53 = K, 
where К = EG — F 2 , substituting these values and observing that 
L + M+N = -LMN, 
the radical of the formula becomes 
V{(••)} = \! (K-2LMNGu + 2WLMNu - WE + ZSFv - Gv J ), 
and the differential equation becomes 
^ (K - 2 LMNGu) = LMN'SF-LMNGv + 2 LMNFu) V{(-)}, 
which can be reduced to the form 
Kdv — LMNFQdu — 2LMNGudv + LMNGvdu _ Hdu — 2LMNFudu 
v [K - <S) 2 E + 2LMN(2) 2 - G) и + 2<5>Fu - Gv 2 } ~ V{(2Lu + 21) (2Mu + 53) (2Nu + S){ '