262 PROFESSOR K. PEARSON AND MR. L. N. G. FILON 



Now 



R = R 44 4- ^i 4 Ri4 + '2 4 Rz4 4- ruRu, 



<i T? I T? I M T? I M "R 

 ' I4"4 "I -"14 T ' 12"24 T^ ' al-t^Sl) 



= r 24 R 44 + r 12 R 14 4- R 24 4- r 32 R 34 . 



Multiply the third of these by r, 4 and the second by r 24 and add, we have 

 = 2r, 4 r S4 R 44 4- r, 4 R 14 4- r l4 R, t 4- r, 8 (r, 4 R 14 4- ?-, 4 R 24 ) 4- ( 4- r, 3 r, 4 ) R, 4 . 



Hence, by the first, 



T i , T> .1) /,. o,, , \ TJ,, i 1) / >f ,, ,, ., , ,, \ 



' 24"14 "T" 'H-"M ^ I 44\'l^ ~~ "'IVyttj ~~ "'1J T ^ 1 34 \' 12' 34 ~~ ' 13' 24 ~~ ' 23' 14/J 



while 



^'l4J- X I4 ~T ''34"24 - ^ v "4* ? '34i l 34- 



By means of these relations, let us get rid of the terms in R 14 and R 24 in (xxxix.) 

 above. 



Re-arranging we have, after some reductions, 



2r,o(l 4,) R + 2r,,R 33 R 44 4- R 34 (r 34 r, 3 (r 13 ?',,7' 23 ) 4- v 34 r 13 (?- 23 7' 12 ?' 13 ) 



4- 91} 1} /"I - . r" Wl - . r 2 ^ 



-(- ^ Lt 34 lV,. u ,. 3t ^ I '12/V 1 '3t) ~ ' V. 



But 



Hence we can divide out by R 34 , and accordingly, 



= {w M (r n r u r. a ) + r 34 r 13 (/- 23 >-,,r 13 ) + 2(1 '/;,)( ri,- al ?- I 



+ 2 w - 'i2 M ( M + H 2 ) +- 2r,,R 34 ] . 

 Noting that 



Rat = v 34 (1 'T>) + r 31 - 41 + r s ,r K )',, (r 31 r 4 , + r 3 ,r 41 ), 



we have, after substitution and rearranging, 



2R,. 13 , 31 (1 r'i,) (1 ?- 3 - 4 ) = (r a r,,r M ) (r, 4 r, 3 r 34 ) + (r, 4 V)( r 



+ (^13 r lt r M ) (r, t r, 2 r 14 ) + (^"u J'la*^) (^3 r M r M ). 

 Or, 



| (''is - 'is'ss) (2i - '2s r si) + (''u - r si'is) (''23 ~ r n'\s) 1 

 13 . I + fas - yi^'st) Q'it ~ ^I^H) + (>'n - VgQOa - nt^t)J 



- ~ ~-~" 



