268] EMANANTS AND PITCH INVARIANTS. 283 



It is easily seen that this equation must reduce to the form 



In fact, seeing 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 conceivable screws in the universe it is 

 obvious that the equation could assume no other form. 



What we 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 a? that is, by 



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 X 1} X. 2 , X 3 , X 4 . We can express 

 the necessary relations by resolving these four forces along each of the four 

 directions successively. Hence 



X, + X, cos (12) + X s cos (13) + X, cos (14) = 0. 



X, cos (21) + X 2 +X, cos (23) + X, cos (24) - 0. 



X l cos (31) + X z cos (32) + X. + Z 4 cos (34) = 0. 



X, cos (41) + X, cos (42) + X s cos (43) + X 4 = 0. 



Eliminating the four forces we have 



1, cos (12), cos (13), cos (14) 



cos (21), 1, cos (23), cos (24) 



I cos (31), cos (32), 1, cos (34) 



cos (41), cos ^42), cos (43), 1 



Thus we learn that every determinant of this type vanishes identically. 



Had we taken live or six forces at the point it would, of course, have been 

 possible in an infinite number of ways to have adjusted five or six forces to 

 equilibrate. Hence it follows that the determinants analogous to that just 

 written, but with five and six rows of elements respectively, are all zero. 



These theorems simplify our expansion of the original harmonic deter 

 minant. In fact, it is plain that the coefficients of x*, of x, and of the 

 absolute term vanish identically. The terms which remain are as follows : 



x ti + Ao? + Ex A + C ic 8 = 0. 



