supplementary to a Paper on the Motion of Fluids. 299 



method is in fact tantamount to taking the sum of the columns 

 P^, 5'Q; but this is not necessary, for Ap^q may be con- 

 sidered as the algebraical sum of the mixtihnear area A P Q 

 B D C, and the mixtilinear area B D C Ap 5-, or (if any line 

 O' C D' be drawn parallel to O C L M D) of A P Q B D' C 

 and BD'C Apq. 



Thus then the mass ■=. f dx {J qdy\fq dy being left in- 

 determinate, and the extremity of x travelled round from C 

 to D, and back again from D to C. 



This will be better expressed by transforming the variable, 

 and summing with respect to some quantity, such as the arc 

 of the curve, which continuously increases, or if we please, 

 with i-espect to 6, the angle subtending any point taken within 

 the curve. 



The mass is then = ± 1^ d^\ {fq dy) . ~-^ V . always 



remembering that no constant need be added ^.ofqdy^ and 

 that the doubtful sign arises from the choice of ways in which 

 6 may be measured round. If the area be not included by 

 one line ; but by several, as for example, by a curve and a 

 right line, the above integral, if broken up into as many parts 

 as there are breaches of continuity, will still apply. 



2. Let us suppose that we have two areas exactly coincid- 

 ing with, and overlapping one another; but the density of the 

 one at (x,y) to be g, and of the other q'. 



Let th^ mass of the first be ti'eated as the sum of columns 

 parallel to Oy, and that of the second as the sum of columns 

 parallel to O x. 



The one will be represented by + / d^ {fp dy) -— 



the other will be represented by + / d^f{p' d x) --^ 

 and the sum of the two, or the joint mass, by 



So long as these two operations are performed separately, 

 the doubtful signs may be preserved in each term, because s 

 need not be travelled round in the same direction for the two 

 summations ; but if we perform the second integration con- 

 jointly for the two masses, their sum 



the mark of interrogation denoting that one or the othcr^ but 



