Mr. A. Caylcy on a Class of Dynamical Problems. 309 



then the required expression of the second Hne will be 



(A0' + H^'. . +P) cd+ (H0' + Bf . . + Q)S^ + . . . 

 which, if we put 



K=i(A0'=+B./,'H.. +2I-I0y + .. +2P9' + 2Q0' + ..). 

 =x(A,B,..H,..P,Q,..p',0',...,l)\ 

 may be more simply represented by 



Only it is to he remarked that A, B, . . H, . . P, Q, . . will in general 

 contain not only 6, <j>, . . , but also the differential coefficients fl' 



(j)', . . , and that in forming the differential coefficients ' — ;, ~, &c., 



_ . (Id' df' 



the quantities 0', (p', . . , in so far as they enter into K, not explicitl}^, 

 but through the coefficients A, &c., must be considered as exetiq^t 

 from differentiation, so that the preceding expression for the second 

 Hne by means of the function K is rather a conventional representa- 

 tion than an actual analytical value. 



Uniting the two lines, and equating to zero, the coefficients of 

 IQ, cip, &c., we obtain finally the equations of motion in the form 



^ ^ _ ^ <7y dK_ 



dtd& dd dd dd'~^' 

 d dT _dT ^(N dK_ 



dt dcp d(j> d<f, df ~ ' 



where the several symbols are to he taken in the significations before 

 explained. 



In the particular problem, let z be measured vertically downwards 

 from the plane of the table, then Z=ff, and repeating for the parti- 

 cular case the investigation ab initio, the general equation of motion is 



KS=-)^ 



dm + 'I^H^da=0. 

 dt dt ^ 



Let s be the length in motion, or, what is the same thing, the z co- 

 ordinate of the lower extremity ; and suppose also that the mass of a 



unit of length is taken equal to unity, we have cz = ?s, -^ = — , 



dm=dz, and the summation or integration with respect to z is from 

 5r=0 to z'^s, whence 



2 ( -4 —g\czdm= {-,-„— (/\^s I,dz= ( -— —a] s2s; 

 \dt- ^J \di- '' ) \dt' ^y 



which is of the form 



it'll -<^^^1\ Is, 

 \dt dsf ds ds J 



if 



i=y-.'s, \=-ffss, 



where the bar is used to denote exemption from differentiation, but 

 ultimately s is to be replaced by s. Considering now the second 



