q/ Edinhurgh, Session 1882 -83. 
165 
We put 
remarking that d and /? represent the projections of a and /3 respect- 
ively on a plane perpendicular to namely, on the plans ^ 77 , we 
get, by comparison 
2 sin u _ sin v 5 
This gives separately 
( 42 ) 
And we must attribute to the second members the signs respectively 
of Srja and of S 77 / 5 , to be determined in each particular case. In 
all cases we have 
[sinM = T. &7ja 
(sin v = TSrj^ . 
So that the signs of Srja and of S 77/3 give the signs of the second 
members of (42). 
The number of arbitrary elements, by which the displacement dp 
of any point p may depend, will also be six in the most general 
solution. Having, namely, 
(43) dp - dp^ + 2rjSrj{p - Pc) , 
and 
(44) dp = dp, + 2&inu.~PY^{p- p^) , 
we may consider p^ as given by 77C which contains three elements, 
two included in and one in C. Then rj being given we have ^ 
by one more arbitary element, because ^ already has to satisfy two 
scalar equations, 
877 ^ = 0 , and C“=-l. 
When 77 and ^ are determined, then ^ will be known by 
Finally, the value of dp^ is of the form 
dp^ = , 
