90 PROCEEDINGS OF SECTION A. 



Now as — 



1 t 'l 

 a, = — s^ (tto + «) = ^ ("0+ «), (see section 8), 



1^.2 ) 



= , a.^ sill ^, -j de sin (b, + 7t) > • 



Again as — ^ 



_ _ 1 _ ' 



2, ' cr, ' 



sin ^, _^ sin /3, ■ 



Va.a,= — - - - af = _^^c„ _l ^) 



similarly — 



and as — 



sin h 



Va,a,= — —^ — 7^ (a„ + k) &c., &c. 



(a^ -{- k) =Uo ~\~ '^^ '\~ ~<^o'< cos a^j/c, 



i 4 



= a^ + — r , e^ + df^ COS 7z. 



WW Mm 



We find, i£— 



that— 



_ sin /?, sin &, sin ^, 



r sm &, , ■) -2 , 2 (■ sin(&, + ^) _ ■) 



^ + B f ^o + - ] — ^ '^ ^ + 2B cos ^ f d,^ 



or in- 

 which must be negative if the machine runs as a motor. 



Now 5„ o-,, S3, o-^, &c., sin i,, sin /S,, sin i^, &c., are all 

 essentially positive, and, therefore, B is so. 



Also a„ and e are essentially positive. 80, in order that the machine 

 may work as a motor h, the phase angle of the applied e.m.f. must have 

 such a value as to make the above expression for T negative. 



pro 



The power supplied by the source e being the mean value of the 



1 ^■ 

 duct of e and x, that is of e and a,, or of e and — *■ (™o + *<) 



s, 



is'|= — ^^ d„ e sin {h — h,) -\- - e k sin h^. (See section 11.) 

 2s, Is, 



= — — s do sin Cli — bj — "' sin b, [ • 

 2s, ( wm ) 



