9 
452 
Tra7isactio7is of the Boyal Society of South Africa. 
Similarly the co- factor of I4I5I6 is found to be 
|{2, + 25 + 26 3, + 85 + 36 
8. The development, however, may also be arranged according to 
products of elements of any other frame-line than the first. Thus, taking 
the third frame-line I3, 2^, 3^, 3^, 36, and seeking the co-factor of 2335, we 
first note that the indices not found in 2385 are 1, 4, 6, and we thereupon 
form the triangular array 
14 16 
46, 
to which we next prefix the line of three elements got from taking 1 along 
with 2 and 5, 4 along with 2 and 5, and 6 along with 2 and 5, the full 
co-factor sought being 
{l.+ L 
14 
26 + 56) 
l6 
4. 
This suggests that the third, and indeed any, frame-line may be made the 
first ; and we readily find that the procedure necessary to effect the trans- 
formation is (1) to delete the third frame-line from the given triangular 
array, and write it again as the first frame-line of a new array, and (2) to 
complete the said new array by appending the undeleted minor of the 
original array, the result being 
II I3 2. 3^ 85 36 j 
1. I4 I5 lej 
2^ 2^ 26 } 
65' 
Since any frame-line may thus occupy any one of the available places, the 
number of different modes of writing the function of the nth degree is 
(7^ + 1) !, the last mode being got from the first by not only reversing the 
order of the frame-lines but also reversing the order of the elements in 
each : for example, the first and last modes of writing the function of the 
4th degree are 
dj 
1^5 ^5 
a. 
a 
3 
a J 
