OF FLUIDS ON THE MOTION OF PENDULUMS. 



[105] 



Note C. Article 50. 



The results mentioned in this article were originally given without demonstration ; but as 

 the mode in which they were obtained is short, and by no means obvious, I have thought, 

 it advisable to add the demonstrations. 



In order that the right-hand members of equations (138) may be perfect differentials, we 



must have 



(Id dw 



do dw 



dy dx 



do dw 



dm dx 



dw dw 



dy dss 



dx dy 



dS 



dx dy 



= 0, 



dw dw 



dx dx 



The equations (c) give 



= 0, 



dw 



= 0. 



di 



do dw" 



T + ST " °» 

 dx dx 



dS dw 

 dy dx 



dw dw 



+ o. 



dx dy 



= 0. 



(a) 

 (6) 

 (<0 



(d) 



dx dy dx 



In the particular case in which 5 = 0, the equations (a), (b), and (d) give 



dw = 0, dw = 0, dw — 0, 



and therefore w', w", and w" are constant as stated in Art. 50. In the general case the equa- 

 tions (a), (6), and (d) give for the differentials of w, w", and w" the following expressions: 



A > dl A dl A \ 



dw = — — dy + — - dx, 

 dx dy 



dS A dS A 



dw = — — dx + — dx, I (e) 



dx dx 



,„ dl dl 



dw = — —dx + — dy. 

 dy dx 



In order that the right-hand members of these equations may be perfect differentials, we must 

 have 



tPS 

 dydx 



= 0, 



dxdx 



= 0, 



<Fl 

 dxdy 



= 0, 



(/) 



d?l d^_ <PS^ <Pl _ d'l d*l _ 



dtf + dx*' ' dx* + dx^~°' dx~ 2 + df~°' 



and therefore 



d^ 

 dx* 



d*l d?8 



Vol. IX. Part II. 



(g) 

 38 



