210 Mr. W. Steadman Aldis. 



In the particular case of n = 0, y l and y 2 may have the values I c (# 

 and K (x) respectively. Hence 



A 



or, by means of the sequence laws, 



Multiplying by x, and putting x = 0, it is easily seen that A = 1, 

 consequently for all values of x t 



The writer has to thank his friend Captain Makgill, E.E., of Waiuku, 

 Auckland, for verifying by this formula most of the results obtained 

 before the writer left New Zealand. For the others he has had to rely 

 on his own verification. 



The formula is an infallible indicator of any mistake in the values of 

 ft or yze, or in the process of multiplication of I (ar) and I-^x) into 

 (logo; E 1). It obviously will not indicate an erroneous value of 

 this last quantity. The values of (log x - E - 1) have been all cal- 

 culated in two different ways, so as to avoid the possibility of mistake, 

 but in order to give the greatest security, a table of the values em- 

 ployed is appended, and the writer hopes that if any mistake is 

 detected, information of it may be sent to him, as it would be a very 

 easy matter to supply the requisite correction to the values of K (a;) 

 and K 1 (ar). 



As a final test of the accuracy of the results, the differences of the 

 column for K (a?) have been calculated up to those of the seventeenth 

 order. Up to this point they present in each set of differences a series 

 of regularly decreasing quantities. In the differences of the eighteenth 

 order this ceases to be the case with regard to the quantities at the 

 lower end of the column. This is due to the accumulation of the effect 

 of residual error in the last figures of the column of values of K (ar). 

 The differences of the seventeenth order at the lower end of the column 

 are quantities consisting of fifteen ciphers followed by six significant 

 figures. Now since 2 20 is greater than a million, it follows that a residual 

 error of four-tenths of a unit in the last figure, in opposite directions 

 in two consecutive values of K O (JC) might possibly, after eighteen 

 differentiations, produce an error of a unit in the sixth place from the 

 end, consequently completely disorganise the sequence of the eighteenth 

 differences which consist only of five figures. That this has actually 

 happened in this case the writer has shown by examining the effect of 

 adding to the values of K O (JC) given in Table I the three additional 



