On the Progress of Accuracy, informing a Plane Surface. 107 



best detector in this particular case, because it is sufficiently sen- 

 sitive, never gets out of order, and indicates almost momentary 

 currents ; it is besides the cheapest instrument that could be used. 



Note. — This method may also with advantage be used for de- 

 tecting bad joints in a telegraph line. It is then only necessary 

 to connect the two ends of the joint to the two terminals of the 

 magneto-electric machine in such a way that the body of the 

 tester acts as a shunt to the joint. 



A joint which offers a resistance of not less than 5 S. units 

 allows a current to pass sufficiently strong to be detected by the 

 tongue ; but if the joint has a resistance of more than 200 S. units, 

 the current passing is strong enough to be felt already by the 

 fingers of the tester. 



XIV. Investigation of the Law of the Progress of Accuracy, in 

 the usual process for forming a Plane Surface. By George 

 Biddell Airy, Astronomer Royal*. 



IN order to form a plane surface, it is usual to take three 

 surfaces (which I shall call A, B, C), and to grind A with 

 B till they fit together, then to grind B with C, then to grind C 

 with A. (I shall call each of these grindings a rub, and the 

 system of three rubs an operation.) And the problem which I 

 propose is, to find the deviation of each surface from a plane, 

 after n operations, expressed as a function of n. 



I shall assume that at each rub the surfaces are worked into 

 perfect contact. Putting A and B for the prominences of special 

 parts of the surfaces of A and B above a mean plane, and put- 

 ting A' and B' for the state to which they are changed after the 

 rub, and remarking that convexity of one corresponds to conca- 

 vity of the other, A' + B' must =0. I shall also assume that 

 equal portions are worked off the two surfaces — that is, that 

 A— A'=B— B'. These equations give 



A'=iA-iB, B'=-iA + iB. 



Considering now the effect produced by the first of all the 

 operations, the expressions for the prominences are as follows : — 



Before rubbing . 



After first rub . 



After second rub. 



After third rub,"] 

 completing the > 

 first operation. J 



A 



+ iA-P 

 + iA-iB 



+ AA-JB-iC 



B 



iA + iB 

 1A + JB-1C 



-JA + JB-'iC 



C 



c 



+ 1A-1B + 1C 

 -JA + JB+*C 



* Communicated by the Author. 



