12 



MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



1.305 A determinant is not changed in value if to each element of a row or 

 column is added the corresponding element of another row or column mul- 

 tiplied by a common factor. 



1.306 If each element of the /th row or column consists of the sum of two 

 or more terms the determinant splits up into the sum of two or more de- 

 terminants having for elements of the /th row or column the separate terms of 

 the Ith row or column of the given determinant. 



1.307 If corresponding elements of two rows or columns of a determinant 

 have a constant *atio the determinant vanishes. 



1.308 If the ratio of the differences of corresponding elements in the p\h and 

 0th rows or columns to the differences of corresponding elements in the rth 

 and sth rows or columns be constant the determinant vanishes. 



1.309 If p rows or columns of a determinant whose elements are rational 

 integral functions of x become equal or proportional when x = h, the determinant 

 is divisible by (x h) p-1 . 



MULTIPLICATION OF DETERMINANTS 



1.320 Two determinants of equal order may be multiplied together by the 

 scheme : 



I ** I X I i, I = 1 c ti | 

 where 



Cij = dubji + dabjz + . . . . . . + d in bj n . 



1.321 If the two determinants to be multiplied are of unequal order the one 

 of lower order can be raised to one of equal order by bordering it; i.e. : 



011 012 01r 



021 022 02r 



0nl 0n2 



1.322 The product of two determinants may be written: 



0nl 



X 



