ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 119 



_ H^H(w,+K) E'E"-A'A" 



,_ Hm,H(/rt, + K) B'B" + A'A" 

 ^'^~~ iB ■ ehc 



,, 1 H\XB"A"+HXia2+K:)B'A' 



y-B' -^ ' 



where the axis of {x) revolves about the azis of z with au angular velocity 



=:m \ 



d loge Hiaj cnogeH(w^+E) "1 



I 



and the axis of .^\ round the axis of z^ with an angular velocity 



_ ?i(A-C) f (? log, Wia^ d log, H(z«, + K) j 

 ~ A 1 cZttj da^ J 



There is also, in the 50th volume of Crelle's Journal, an elaborate memoir 

 on the application of the functions Q to the solution of the problem of ascer- 

 taining the motion of the spherical pendulum, by Dumas. 



Section 4. — It will be interesting, in writing on elliptic functions in a 

 country so dependent for its greatness, under Providence, upon its manu- 

 facturing skill as this, to show that these integrals are capable of a direct 

 application to machinery. A remarkable example of this is given by Canon 

 Moseley in his ' Mechanics.' 



The quantity of work done by a pressure P acting through a space S, 

 where P and S are constant, is taken to be equal to PS. Hence if P is 



variable, the work done is equal to 1 Pf?S, or half the vis viva accumulated 



while the work is being done. Canon Moseley then shows that in any 

 machine, if Uj is the work done at its moving point through the space S, 

 Uj the work yielded at the working points, Uj and U^ are connected together 

 by an equation of the form Uj=AU2 4-BS, where A and B are constants 

 dependent for their value upon the construction of the machine, — -that is to 

 say, upon the dimensions and combination of its parts, theii* weights, and 

 the coefficients of friction at the various rubbing-sm-faces. Upon this prin- 

 ciple Canon Moseley works out his theory, and the above equation is applied 

 to the wheel and axle, to pulleys combined in different ways, to toothed 

 wheels, and to aU the component parts of machinery, affording in many cases, 

 and especially with regard to toothed wheels, results of great interest and 

 beauty. 



In the case of the capstan, the above equation leads to an elliptic function. 



Let a^ be the length of the lever turning the capstan measured from the 

 axis; 



a^ the length of the perpendicular upon the rope supposed to act in a con- 

 stant direction ; 



T the tension of the rope ; 



Uj the work done by the pressure applied to the extremity of the lever 

 always perpendicular to its direction ; 



