6 GENERAL INTEGRALS OF PLANETARY MOTION. 



d% /OooOio I CColdil I «02ai2 I _|_ «0»»«-m \ ^^ 



/OioG^o I cxii«n I ai2ai2 , _ . . . _L "i"^'"' \ ^^ 

 ' \ mo Wi "'" mg "i« ' oi^i 



'V ?7?o ??ii ■ m^ "•" "^ m,, I d^n 



Tn order that this equation may reduce to the canonical form 



d^i d£l 



it is necessary and sufficient that the expressions 



Too TOj m2 ??■(„ 



should vanish whenever i is dilFerent from j, and should reduce to unity w^henever 

 i=j. In other words, it is necessary and sufficient that the coefficients a should 

 be so chosen that the {n -{-If quantities 



Otpo aoi Ctpn 



•i/m,o' -|/mi i/m^ 



■ \ \ \ (8) 



|/mo' i/toi -j/*",,' 



should form an orthogonal system. The first line of coefficients is already deter- 

 mined by the equation (6), the coefficient c excepted, which is to be determined 

 by the condition 



+ ^+ +""-^ = 1, 



Too WJi ™„ 



C-oo I CXoi 



or, from (6) 



»»o + "'i +••••+ '»)» == «"' 

 which gives 



c = i/m, 



putting m for the sum of the masses of the entire system of bodies. Having thus 



TOj 



the orthogonal system (8) becomes 



I/toq t /^l t/^» 



|/??< ' -(/??« ' -|/to 



-j/TOo' /TOi' -j/to„ 



-j/TOo' -j/TOi' |/to„° 



