Arbitrary Constants in Mechanical Problems. 255 



By similar steps to those which I employed in vol. xi. p. 494, 

 it is easy to show that 



^^^ dt=(b,a)db-\-{c,a) dc^(e,a)de + 8ic. (I.) 



R not containing 45', y, or 6' explicitly, so that 



^^^-n iL?-o ^-O 



rfi? ^^ ^Zi2 6^4) ,, , dR J^^ ,, , rfi? d$ - 

 -- — ^^= -j~ -=^ dt+ -YT -J- d^^-^ir -7—^^ 

 da d (p da d^f da du da 



(b, a) denotes the quantity 



dd> ds ds d (p , o a • 



rrr i -jt "T^ + &c. Again, 



db da db da ^ ' 



dU_^dJl^dj> dU^d^ ^ &c 

 da d<p da ' d<p' da 



"" dt Ua"^ d<p' da "^^^* 



where I have omitted to write down the similar terms with re- 

 spect to t]/ and 5. 



d,il d, dU 



d' [7 __ dt jH __ ds ^^J> a<p' d<p' 



dadb "~ dl) da dt dadb db da' 



^ d<^. dadb ^^^' 

 Similarly, by interchanging the letters a and b, 

 d'£s ^ .dU 



rf dt d (p ds d^<p d<p' d<p' 



dbda ~" da db ^ dt dbda da db 



d <p dbda 

 Subtracting this equation from the last, 



