548 ' REPORT— 1881. 



The most general set of forces acting upon a system can also be represented by 

 a series of wrenches on the screws of a screw-chain. The screw-chain is thus 

 symmetrically related to the most general group of forces and to the most general 

 form of displacement. 



The theory is extensive, but one illustration may be given. If the forces 

 act impulsively, the system commences to twist about an instantaneous screw- 

 chain. Generally the impulsive screw-cliain and the instantaneous .screw-chain are 

 diiferent. It will, however, sometimes happen that the two coincide. The number 

 of cases where this occurs is equal to the number of degrees of freedom which the 

 system posses.ses. 



4. On a Properhj of a small Geodesic Triangle on any surface. By Professor 



H. J. S. Smith, M.A., F.B,.8. 



5. On the General Analogy between the formulce of singly and doubly Periodic 

 Functions. ' By J. W. L. Glaisher, M.A., F.B.8. 



■ The object of the paper is to consider the general groups of formulae in Elliptic 

 Functions, the existence of which is indicated by form ulfe involving trigonometrical 

 functions. As we should expect, the former are much more complete, and exhibit 

 greater regularity, than the latter. 



Change of the argument. — In Trigonometry tliere are six primary functions, 

 sin u, cos u, "tan n, cot u, sec tt, cosec ti, of which two are even and four are uneven. 

 These six functions are divisible into three pairs, sin u and cos u, tan u and cos ti, 

 sec M and cosec u, which are such that if we select from them any triad, one being 

 taken from each pair, the other three functions may be derived from them by a 

 change in the argument, that is by the substitution of ^tt + m for u. The num- 

 ber of such triads is eight, the total number of triads that can be formed from the 

 six functions being twenty. 



In Elliptic Functions there are twelve primary functions, sum, cdm, dnu, 

 1 1 1 an M sn M en M en j( dn « dn M 



; •, -q — , , J — ) — ) 1 — ! » • — » of which SIX are even 



snu cnu dnw cnu dnw snu anu snw cnw 



and six uneven. These twelve functions are divisible into three groups of four 

 each — viz., 



snu, > ■ — ) 



' sn M en ?* 



dn u en u 

 dn u 



dnw 1 sn u 

 sn M en u dn u 



en u sail 1 

 sn u en u dn u 



which are such that if we select from them any triad, one being taken from each 

 group, then (omitting the constant multipliers -r, k', &c.), the other nine functions 



may be derived from this triad by changes in the argument, that is, by the substi- 

 tution of K + M, iK' + M and K + iK' + M for u. The number of such triads is 

 sixty-four, the total number of triads that can be formed from the twelve functions 

 beinc 220, If we consider only the real substitution — viz., that of K + u for u — the 



., , , , . . en M sn M 



twelve functions are divisible into the six pairs sn u and t— , en n and j^j dn u 



] 1 dn?< 1 Anu mu , en m , ., . 



and T — I and ^ » and 1 ■ — and which are such that 



dn u sn ?* en u en u sn u en u sn u 



if we select from them any six functions, one being taken from each pair, the 



other six may be derived from them by this substitution. 



Integration. — We have 



