460 



PRACTICAL MATHEMATICS 



numbers, the investigation would be easy. Find if there is approxi- 

 mately such a law within the limits r = 0-5 and r = 0-7, and what 

 is the maximum error in making such an assumption. (B. of E., 

 1908.) 



(5) When a shaft fails under the combined action of a bending 

 moment M and a twisting moment T, according to what is called 

 the internal friction hypothesis, M + a 



+ T 2 ought to be con- 



stant where a is a constant. Test if this is so, using the following 

 numbers which have been published. Considerable errors in the 

 observations must be expected. (B. of E., 1910.) 



(6) In the following table C denotes the radio-activity of a sub- 

 stance, t hours after the observations were commenced. There is 



-ir\ 



reason for believing that -^- = aC where a is a constant. Try if this 



CLv 



is so, and if so, find the most probable value of a. (B. of E., 1911.) 



(7) The following values of x and y were observed in a laboratory 

 and theory suggested that there might be a law y = ax + b Iog 10 x. 

 There are errors of observation. Try if there is such a law, and 

 if so, find the most probable values of a and b. (B. of E., 1912.) 



(8) It is thought that the following observed values of x and y 

 follow the law y = A^ x . There are errors of observation. Test 

 if such a law is probably true, and if so, find the values of A and b. 

 (B. of E., 1913.) 



