408 Mr. Challis on the general Equations 



ds 

 not vary, dw = gdt ; therefore ^ =a + w, and the velocity of pro- 

 pagation is exactly a. This result might have been deduced 

 from the general formula, w = a hyp. log. f + (p {t), in Art. 5. 

 The same velocity of propagation will be found by employing 

 the other integral. 



8. I return now to the equation (p) Art. 5. Let P be the 

 force obtained by resolving the impressed forces in the direction 

 of r, and let it be either constant or a function of r. Then 



'^__i^ i?!^__L_ ^- f^ + p')~-^*^ =0 (M) 



dr' a- - a.^ ■ drdt a"- or' dt- \ r ^ a" - a." 



«= hyp.log.p-/Prfr+^+|=o (N). 



By treating (M) according to the method of Monge, the following 

 two sets of equations are obtained : 



dr — adt— wdt = 



wdr 



(«)= 



I dd) /2a' „>Y wdr 

 adw - todu> -rf-"77 + ( — +P) ^7T— = 

 dt \ r y a + w 



dr + adt — ix)dt = o j 



dd) /2a- , r,\ wdr ( ^P^' 



adw + wdw + -^ + ( — +P) =0 



at V r / a— w ' 



Now multiplying the first of the equations (a) by P, inte- 

 grating the two, and adding the integrals, we shall have, because 



dr 



= dt. 



a + w 



w' 



aw — 



# ^ i-Drl^ J. on"- r^ 



dt 



+/Pdr + 2fr f— - af, Pdt = c+c'^c +J\ (c) =,/(c). 



dv 

 So from the equations (/3), because -^ = - dt, 



