- combined with the force Ab, consti- 
Prof. Twining on the Parallelogro 
to AO. Make OAO’ ons Oo BAO; and, by what is already sup- 
posed, the effect of the force 
and its residual force will be bE normal to AO’. Then AB is 
the resultant of the three forces Ab, bE, EB. But, since the force 
EB is normal to AO, its effect normal to AO’, by what is sup- 
posed, must be eB; also in the direction AO’, it must be Ee or 
bE’. The first, combined with the force bE, constitutes the 
force E’B, normal to AO’, and the last, 
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 Steet 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 O”AB is 
represented bythe 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 2’ 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 »z with its own direction, 
Would be represented, in intensity, by the cosine of n2’. If, 
then, DAB and CAB are commensurable, the former may have 
nz fie 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. xzv1, No. 2.—Jan.-March, 1844. 42 
