Prof. Twining on the Parallelogram of Forces. 329 



to AO. Make OAO / equal to BAO ; and, by what is already sup- 

 posed, the effect of the force AE, in the direction AO' will be Ab, 

 and its residual force will be Z»E normal to AO'. Then AB is 

 the resultant of the three forces Ab, 6E, EB. But, since the force 

 EB is normal to AO, its effect normal to AO', by what is sup- 

 posed, must be cB ; also in the direction AO', it must be Ec or 

 6E'. The first, combined with the force 6E, constitutes the 

 force E'B, normal to AO', and the last, 

 combined with the force Ab, consti- 

 tutes the force AE' ; which therefore 

 is the effect of AB in the direction AO', 

 and its residual effect normal to AO', is 

 E'B. In like manner may it be seen 

 that, if O'AO" be taken equal to BAO, 

 the effect of AB in the direction AO" is AE", and its residual ef- 

 fect, normal to AO", E"B; and, in general, that the effect of AB, 

 at any multiple of BAO, would be represented, in intensity, by 

 the cosine, to radius AB, of the same multiple of BAE, and, of 

 course, its residual effect by the sine of the same. Conversely, 

 also, it is evident, that if the effect of AB at any angle 0"AB is 

 represented by the cosine of another angle E"AB, then will the 

 effect of AB, at any exact part, or measure of the first, be repre- 

 sented by the cosine of the same part of the second ; and, of 

 course, the same, mutatis mutandis, of the residual forces and 

 sines. 



To apply this to the point in hand, — let CAB (fig. 4) be suppo- 

 sed to vary from the corresponding angle of the diagonal of the 

 parallelogram CAD, if completed ; that is, from the angle which 

 that diagonal would make with AC. Take z any part or meas- 

 ure of the resultant angle, and z' the same part of the corres- 

 ponding diagonal angle. Take nz, nz' multiples of these by an 

 integral number. Then, by what has been proved, it appears 

 that the effect of AB, at the angle nz with its own direction, 

 would be represented, in intensity, by the cosine of nz'. If, 

 then, DAB and CAB are commensurable, the former may have 

 nz for its equal, and therefore nz, or DAB, must vary the same 

 way, in respect to excess or defect, from the corresponding diag- 

 onal angle, as DAB from its corresponding diagonal angle ; so 

 that, in this case, the two together would constitute DAC less or 



Vol. xlti, No. 2.— Jan .-March, 1844. 42 



