of Edinh^irgli, Session 1882-83. 
155 
Should da and d,p require to be parallel, then we would, d fortiori, 
be entitled to give to and to Vg' the same direction, considering 
that both would be comprised in one and the same plane, and the 
relations (9) would again determine u and v. 
Let us now introduce the expressions (11) into Z, remembering 
that we have now by (9) 
(12) da = ^]Nla . sm u, dp = V . 
Moreover, as 
we may write 
We have now by (12) 
S. qYtP = 0 
Z = S . idadji . 
2 sin u sin v 
The relations (10) become 
(13) S^c?a = 0, ^tdp = 0. 
This gives 
±UV. dadp 
8(dadp = =F TY.dadp . 
The equation Z = 0 becomes then finally 
T.Vfa 'j)) T 
2 sin ti sin v \ 
We have therefore two solutions for the equation (2), 
I. either T.Y{q~f) = 0 , namely ±q=p; 
II. or TY{dadp) = 0, 
namely, da and dp to be parallel to one another. 
We omit such particular cases for which both conditions are 
satisfied at the same time. 
§ 2 . 
For the discussion of these two solutions we will refer to a fixed 
origin 0, the position of the points which are invariably linked 
to the system. Let p^, p^, &c., p, designate the vectors of these 
points before their displacements, and let p\, p &c., p be the 
vectors of respectively the same points after their finite displace- 
