376 Proceedings of the Royal Irish Acadejmj, 
I already pointed out that this equation must necessarily reduce 
to the form 
= 0. 
In fact, seeing that it expresses the solution of the problem of 
finding a screw of maximum pitch, and that the choice may be made 
from a system of the sixth order, that is to say, from all the conceiv- 
able screws in the universe it is obvious that the equation could 
assume no other form. 
What I now propose to study is the manner in which the necessary 
evanescence of the several coefficients is provided for. After the 
equation has been expanded we shall suppose that each term is 
divided by the coefficient of that is, \)J PiPzPzPi. P^Pq- 
The work is much simplified by a few simple theorems which are 
I doubt not, well known, but with which I was not acquainted 
until they came to light in this investigation. However, the mode 
of proof which I have adopted, being of a mechanical nature, may 
be worthy of record. 
From any point draw a pencil of rays parallel to the six screws. 
On four of these rays 1, 2, 3, 4, we can assign four forces which 
equilibrate at the point. Let these magnitudes be Xi, X2, Xj, X4. 
We can express the necessary relations by resolving these four forces 
along each of the four directions successively. Hence 
Xi + X3Cos(12) + X3 cos(13) + X4 cos (14) = 0. 
Xi cos (21) + X2 + X3 cos (23) + X4 cos (24) = 0. 
Xi cos (31) + Xo cos (32) + X3 + X4 cos (34) = 0. 
Xi cos (41) + X2 cos (42) + X3 cos (43) + X4 =0. 
Eliminating the four forces we have 
1, cos (12), cos (13), cos (14) 
cos (21), 1, cos (23), cos (24) 
cos (31), cos (32), 1, cos (34) " ^' 
cos (41), cos (42), cos (43), 1 
Thus we learn that every determinant of this type vanishes identi- 
cally. 
Had we taken five or six forces at the point it would, of course, 
have been possible in an infinite number of ways to have adjusted five or 
