176 Mr CHALLIS's RESEARCHES IN THE THEORY 



of the quantities which the function F' involves. For the equation 

 ^j% + -T-? = 0, is also satisfied by the following, 



((> = F {{a^- X cos d - y m\ Q) + {^ + xsm9 +y cosO) ^/^^} 



+y {(" + a; cos 0-y sin 0) - (/3 + a; sin f y cos 9) V~^\ '■> 



and this analytical circumstance has its interpretation in reference to 

 the motion of the fluid. By supposing the function f to be the same 

 as F, and giving to F' the same form as before, we shall find, 



' d^ _ 2 C(.r + g cos g + /3 sin 9) 



dx ~ (a + xcos9 — i/ sin 9y + (13 + x sin 9 + y cos 9)- 



d^ 2C(y + /3cosg-as in 9) 



dy {a -\- X cos 9 — y sin 9'f + (/3 + a; sin + y cos 9Y 



/d^y /d^Y 4C^ 



\dx ) \dy) ~ {a + x cos9-y sin9y + (^ + x sm9 + y cos9y 

 Or, if a cos 9-\-fisin9= —a, and /3 cos 9 — a sin 9= —b, 



d(f> 2C(x-a) 



dx (x — ay + {y — bf ' 



^ or V- ^^(y-*) 

 dy ^x-ay + iy-by 



Vu' + v^ = 



y/ix-af + iy-by' 



This shews that the velocity is directed to the point whose co-ordinates 

 are a, b, and varies inversely as the distance from it. And as we have 

 arrived at this result without considering any circumstances under which 

 the fluid was caused to move, the inference to be drawn is, that such 

 is the general character of the motion. Nothing forbids our considering 

 C, a, and b, functions depending on the time and the given conditions 

 of motion in any proposed problem. Also if at a given instant, a line 

 commencing at any point, be drawn continually in the direction of the 

 motions of the particles through which it passes, C, a, and b, may be 



