412 THE THEORY OF SCREWS. [375, 



and as #/, a , &c., are indefinitely small, this reduces to 



2*1 = 0, 

 where 2\ is a homogeneous function of lt 2 , ... B n in the second degree. 



For the study of the permanent screws we have, therefore, n equations 

 of the second degree in the co-ordinates of the instantaneous screw-chain, 

 and any screw-chain will be permanent if its co-ordinates render the several 

 differential coefficients zero. We may write the necessary conditions that 

 have to be fulfilled, as follows : 



Let us denote the several differential coefficients of T with respect to the 

 variables by I, II, III, &c. Then the emanant identity is 



1 i + 2 n + 3 ni + ...=o, 



and we may develop any single expression, such as III, in the following 

 form : 



III = III! A 2 + III 22 2 2 + III^s 2 + 2111,-M + . . . 2III M 4 . 

 As the emanant is to vanish identically, we must have the coefficients of 

 the several terms, such as Of, Q?Q. ly A0Ai & c -&amp;gt; all zero, the result being 

 three types of equation 



In = 0, I-B+ II 12 = 0, I 23 +II 1 3 + IIIi 2 = &amp;gt; 



1122 = 0, II U + I 12 =0, &C., 



11133 = 0, II B + III B = &amp;gt; &C. f 



IV^ = 0, &c., &c. 



&c. 

 Of the first of these classes of equations, I n = 0, there are n, of the second 



A e^ +u- A n(n-l)(n-2) . w(n+l)(n+2) 



there are n (n - 1), and of the third, ^^ , in all, - i ^ 5 . 



1 . L . o 1 . Z 



376. Another identical equation. 



Let T be the kinetic energy of a perfectly free rigid body twisting for 

 the moment around a screw 6. It is obvious that T will be a function of 

 the six co-ordinates, #/, . . . # 6 , which express the position of the body, and 

 also of 0j, ... 6&amp;gt; the co-ordinates of the twist velocity, 



T=f(0 l ,...0. t ft, ...A). 



We may now make a further application of the principle employed in 

 302. The kinetic energy will be unaltered if the motion of the body be 

 arrested, and if, after having received a displacement by a twist of amplitude 

 e about a screw of any pitch on the same axis as the instantaneous screw, 

 the body be again set in motion about the original screw with the original 

 twist velocity. This obvious property is now to be stated analytically. 



