ON A CLASS OF DYNAMICAL PROBLEMS. 
9 
225] 
Considering next the second term, or 
we have here 
8£ = a 86 + b 8(f> + ..., 
8r) = a' 86 + V 8(f) + ..., 
8Ç=a"86 + b"8(f> + ... , 
where a, b, a', &c., are functions of the variables 6, <f>, &c., and of the constant para 
meters which determine the particular particle dfx. The virtual velocities or increments 
86, 8(f>, &c., are absolutely arbitrary, and if we replace them by d6, d(f, &c., the actual 
increments of 6, </>, &c. in the interval dt during the motion, then 8%, 8i7, 8£ will 
become ~ dt, — dt, ^5 dt, in the sense before attributed to ~ ^. 
dt dt dt dt dt dt 
The particle d/x will contain dt as a factor, and the other factor will contain the 
differentials, or (as the case may be) products of differentials, of the constant parameters 
which determine the particular particle d/x. We have thus the means of expressing 
the second line in the proper form; and if we write 
2 (a 2 + a' 2 + a” 2 ) d/x = Adt, 
2(6 2 +b' 2 +b" 2 ) d/x = Bdt, 
2 (ab 4- a'b'+ a"b") d/x = Hdt, 
2 (au + a'v + a"w ) d/x = — Pdt, 
2 (bu + b'v + b"w ) d/x = — Qdt, 
then the required expression of the second line will be 
(A6' + H(f)'... + P) 86 + (Hff + Bcf>'... + Q) 8cf) + ..., 
which, if we put 
K = \(A 6' 2 + B(f)' 2 + ... + 2 H6'(f)' + ... + 2 P6' + 2 Q(f)'+...), 
= h(A, B,...H,...P, Q,...\6\ I) 2 , 
may be more simply represented by 
dK 
d6' 
86 + 
dK 
d(f) 
-, 8(f + 
only it is to be remarked that A, B,...H,...P, Q, ... 
only 6, (f>, ... , but also the differential coefficients 6', </>', .. 
differential coefficients &c., the quantities 6', </>', 
will in general contain not 
.. , and that in forming the 
... , in so far as they enter 
C. IV. 
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