November 19, 1891] 



NA TURE 



phase is always nought. This may be easily proved thus : 

 ihe values of three such currents are given at any moment 

 by projecting on a stationary line, POO (Fig. 21), the three 

 equal limbs of the three-legged figure Oa, O^, Oy, as it 

 rotates with uniform velocity round the point O. Oa, 0(3, 

 Oy, are, therefore, the maximum values of the three cur- 



55 



struction be made for Fig. 21, f and O will always coin- 

 cide, therefore the sum of the projections of Oa, 0/3, and 

 Oy, Fig. 21, must be always nought. 



Fig. 23 shows three curves, I, II, III, drawn so as to 

 give the values at any moment of three harmonic alter- 

 nating currents each of the same altitude, H, and periodic 



Fig. 18 —Currents differing by 60' in phase, and represented in direction and magnitude by the direction and length of the arrows. 



rents, and the actual values of the currents for the 

 position of the figure shown are OA, O B, OC, respectively, 

 corresponding in relative magnitude with the lengths of 

 the arrows which are attached to A, B, C. in Figs. 18 and 

 20, and in direction with the arrows attached to the latter 

 figure, on the assumption that a current is regarded as 



time, but differing by 120° in phase, and it is seen that 

 the sum of the three ordinates— that is, the ordmate of the 

 top curve— varies from H + 2H sin 30°, when the time 

 equals /, to 2H sin 60°, when the time equals /', so thatjhe 

 ordinate of the summation curve varies from 2H to \''3H, 

 corresponding with a variation of 14 per cent. But this 



Fig. 19.— Three harmonic alternating currjnts of the same period and maximum altitude, but differing by 60° in phase. 



positive when it circulates round the iron ring in such a 

 direction as to tend to send a north pole counter-clockwise 

 round the iron ring. Now the sum of the projections on 

 POQ of any three lines Oa, 00, Oy (Fig. 22) is simply the 

 projection of Of found by drawing at and ff parallel and 

 equal to 00 and Cy respectively. But if such a con- 



NO. I 151, VOL. 45] 



is exactly the variation that we obtained in Fig. 19, 

 hence if there be twice as many convolutions in each 

 of the three coils of Fig. 20, as in each of the six coils 

 of Fig. 18— that is, the same total number of coils in the 

 whole ring— and if the three equal harmonic alternating 

 currents differing by 120'' in phase have each the same 



