332 



DR THOMAS MUIR ON THE 



and on partitioning this into two the first is seen to be 



= (1) V 'LMN + C ^NPQ + B 7MRP + A X /LQE) 

 . (D JLMN + C jWPQ - B JMK&-A JLQR) 

 . (D VLMN-C ^/NPQ + B VMEP- A JLQR) 

 . (D VLMN-C j¥¥Q-B /MRP + A JLQB) , 



and the second to be 



D VLMN C JNPQ B JMRP 



C VFPQ D jTMN A JLQR 



B VMBP A VLQE D JJMN 



AV'LQR BJMRP C JWFQ 



and therefore 



D VLMN .C VNPQ .B JMEt 

 C JKPQ B» VLMN.A JLQR 

 B VMPP .A 7LQR .D ^LMN 

 A VLQR .B VMRP .C JNVQ 



LMNPQR, 



= ( C JNPQ . A JLQH - D JLMN . B ^MRP) 

 (A JLQR . B VMEP - C VNPQ . D ^LMN) 

 (B 7MRP. C VNPQ -A JLQR . D JJMN) 



= 2(CAQ - DBM) ( ABR - CDN) (BCP - ADL) . 



iLMNPQR , 



11. Were it not for the divisor LMNPQR attached to the second determinant in 

 the preceding section, the full determinant would be a function of only four variables, 

 viz. : — 



AVLQR, 



BJMRP, 



(VNPQ, 



DJIMN; 

 and as a matter of fact the final expansion of it may be written 



2(A VLQR)*- 22(AVLQR)2(B JMRVf 



+ 8(A ^LQR . B VMRP . C JXPQ . D ^LMN) 



+ 



42(A VLQR^B 7MRP) 2 (C ^NPQ) 2 - 42(A VLQR) 3 .B ^MRP.C ^/NPQ.D Jim 



LMNPQR 



12. Standing in close connection with the subject-matter of the preceding sections— 

 the connection of general with particular — is the problem of clearing the equation 



x -f h Jbc + k Jca + I Jab = 



of root-signs, or of transforming a fraction of which x + h^/bc + ktjca + ljab is the 

 denominator into one having its denominator rational. Viewing the matter in either 

 way we reach the result 



(r + hjbc + h Jca + 1 J~ab) (x + h Jbc -I Jca - 1 Jab) (x - h Jbc + k J~ca - 1 Jab) (x - h Jbc - k Ja + 1 s 



or 



x* + hW + k*c 2 a 2 + / 4 a 2 5 2 

 -2xX?Mc + k*ca + l 2 ab) 



- 2abc(h 2 k*c + WPa + lVi 2 b) 



— Sxhklabc . 



