DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
Ill 
for, referring to the figure, and using the theorems (40.), we see that it is equivalent to 
P= cos _1 (— cos 0)+— ^ C ° SZ cos~ 1 (cos(I+J)) 
_cov_cos« cos — ,(_ C 0 S ( I _J)) + K; 
where K is put for the arbitrary function. Now the expression for ud6 (from which 
this is derived) shows that the three terms in the above value of P must be so inter- 
preted that the differential coefficient of the first (with respect to 0) shall be positive, 
and those of the two others negative. These conditions will be satisfied by taking* 
±P=T _ 0+ £2!i|f£l! ( I + j ) 
_cos/-cos.^_^_j^ +K 
(in which the upper sign is to be taken when © is between o and r, and the under 
sign when 0 is >tt). 
Hence, assuming the arbitrary K so as to destroy the constant part of the expres- 
sion, we have, without ambiguity, for all values of the variables, 
— I cosj — J cos«), 
so that, finally, 
V=&{('4/ — I) cosj+(<p— J) cos z + 0} (48.) 
It will be observed that without attention to the proper interpretation of ambiguous 
symbols, a completely erroneous expression for V might have been obtained. 
44. The final equations will be (art. 19.) 
*I =t+T -^-=« — =S 
dh ' ’ d cos i ’ d cos j 
r, a, (3 being three new arbitrary constants, namely, the elements conjugate respect- 
ively to h, cos i, cos j. 
In performing the differentiations, it is to be remembered that I, J, 0 do not con- 
tain A; and that, by the equations (42.), (43.), art. 39, the terms arising from the 
differentiation of I, J, 0 with respect to i and j, disappear identically, so that these 
functions may be considered as exempt from differentiation. Also we have 
dk k dk 
dh 2 li dc.o%i ’ 
dk (C— A)^ 3 cos < ; 
dcosj 2A.Ch 
* In the figure, as 0 diminishes ( i andy remaining constant) 9 increases, 1 + J increases, and I — J increases or 
• j 
I— J 0 sm — ~ 
diminishes according as j ^ i, since tan — — = tan — . 2 
^ J 
sm 
J + i 
