154 Froceedings of the Boyal Society 
contained in j) and q taken together, and, if we introduce the condition 
(7) UVi^ = UV^ = ^, 
we reduce to four the number of the independent elements con- 
tained in p and q. By this means we do not reduce the number of 
the independent elements on which a! and depend. 
Considering the question in its geometrical aspect, we represent 
the finite displacements by da and dpf putting 
(8) a! — a — da, /3' = /3 = d/3, 
and applying (5) to the expressions (3), we get 
(9) , da = 2pY(p . a) , d/3 = 2qV{Yq . /3) . 
Hence 
(10) S.daYp = 0, S.S^Yq = 0. 
The directions of Yj) and Yq are thus limited to planes respectively 
perpendicular to da and to dp, these planes passing through the 
origin 0^ of a and /?. Moreover these planes contain a definite 
point each. Namely, if we put (1) under the form 
{a + daf^a^= 0 , 
we deduce 
S(a + \do)da = 0 ) 
likewise we get 
S(;g + ^dp)dp = 0. 
If then a and p, da and dp he given, the axis of p and q may still 
remain of an indeterminate position each in its plane, perpendicular 
to da, and containing a -H ^da , or perpendicular to dp, and containing 
the point p + \dp . 
But these two planes generally intersect each other, and if ^ 
represents the unit-vector in the direction of the line of inter- 
section, we may take this direction for those of Yp and Yq in com- 
mon without affecting the values of da and dp. 
We define therefore the expressions of j?, q by 
U = cos ?^ + ^sin u, 
^ \q = cos v + ^ sin v , 
and we determine the angles u and v by the relations (9); inversely 
if we give ourselves u, v as data, the expressions (9) will represent 
any values of da and dp compatible with the two first conditions (1). 
* The characteristic d representing finite differences throughout this paper. 
