A Treatise on the Theory of Screws

260, 261] EMAN ANTS AND PITCH INVARIANTS. 275 We have now to form the co-ordinates of the resulting Dyname, or its components when decomposed along the six screws of reference. The first Dyname has a component of intensity on the first screw; and as the second Dyname has a component yßlt it follows that the sum of these two must be the component of the resultant. Thus we have for the co-ordinates of the resultant Dyname the expressions ®«i + yßL, ... xae + yße. 261. Emanants. Let us suppose that without in any particular altering either of the Dynames a and ß we make a complete change of the six screws of reference. Let the co-ordinates of a with regard to these new screws be ... X8, and those of ß be y1, ... y6. Precisely the same argument as has just been used will show that the composition of the Dynames xa! and yß' will produce a Dyname whose co-ordinates are x\x + yyx, ... xX6 + yye. We thus see that the Dyname defined by the co-ordinates xa1 + yß1, ... xae + yße, referred to the first group of reference screws is absolutely the same Dyname as that defined by the co-ordinates «Xj + y/^, ... #X6 + y/j6 referred to the second group of reference screws, and that this must remain true for every value of x and y. In general, let 0lt ... 9.. denote the co-ordinates of a Dyname in the first system, and <f>1, ... c/>6 denote those of the same Dyname in the second system. 161/(0!, ... 0e) denote any homogeneous function of the first Dyname, and let </>6) be the same function transformed to the other screws of refer- ence. Then we have as an identical equation which must be satisfied whenever the Dyname de- fined by ... f)e is the same as that defined by </>i, ... <j>6. We must there- fore have /(««J + yß1>... xa6 + yß6) = F(xX1 + yylt... xXa + yy6). These expressions being homogeneous, they may each be developed in ascending powers of . But as the identity must subsist for every value of oc this ratio, we must have the coefficients of the various powers equal on both sides. The expression of this identity gives us a series of equations which are all included in the form * * See Proceedings Roy. Irish Acad., Ser. n. Vol. in.; Science, p. 661 (1882). 18-2