We easily obtain, where \ 0 > Po> v o, k 0 are written for A., p, v, k, to denote this particular 
supposition, 
K 0 Ap = 2 (v Q \ 0 — p 0 ) k, 
K 0 Bq = 2 (p 0 v 0 + A, 0 ) k, 
K 0 Cr = (1 + v<? — V — p 0 2 ) k ; 
whence, and from /¿ 0 = 1 + A 0 2 + p 2 + v 2 , K 0 Cr = (2 + 2r 0 2 — k 0 ) k, we obtain 
(1 + 7' 0 2 ) Ap 
k + Cr ’ V ° X ° ^ k+Cr 
Hence, writing h = Ap? + Bqp + CrP, the equation 
dv o 1 
I x _(1 +V)Si 
/v,+x ° * + 
d</> 
reduces itself to 
or, integrating, 
{(v 0 \ 0 — p 0 ) p + (p 0 v 0 + \ 0 ) q + (1 + v~) r} 
4 dv n 
1 + v 0 2 d(f> (k + Cr) pqr ’ 
4 tan -1 p 0 — j , 
(h + Icr) dcf) 
{k + Cr) pqr 
The integral takes rather a simpler form if p, q, </> be considered functions of r, 
and becomes 
2 tan -1 j' 0 = 
h + kr 
C V (AB) dr 
k+Cr V[i k 2 -Bh + {B-C)Cr 2 ) [Ah — k 2 + (C—A) Cr 2 }] ’ 
and then, v 0 being determined, \ 0 , p 0 are given by the equations 
„ v 0 Ap + Bq v^Bq — Ap 
°~~k + Cr ’ /i °" k + Cr ’ 
Hence l, m, n, denoting arbitrary constants, the general values of A, p, v, are given 
by the equations 
P 0 = 1 — ZA 0 - mp 0 — nv 0 , 
P 0 \ = l + \ 0 + m v 0 — np 0 , 
P 0 p = m+ p 0 +n\ 0 —lv 0 , 
P 0 v=n+ v 0 + lp 0 — m\ 0 . 
In a following paper I propose to develope the formulae for the variations oi 
the arbitrary constants p lt q X} r lt l, m, n, when the terms involving V are taken into 
account.