xviii COMITTA Tm\ RULES, 



Observe that the partial products beyond the place standing over the tifth 

 place of the result in each multiplication aiv useless. Hence the obvious saving 

 of labor in the shortened process, whieh is also more compact. The process is 

 easily understood by inspection of the example. Multiply first by the tirst left- 

 hand figure of the multiplier. If the resulting partial product has one more 

 place than is desired in the result, then drop the last ligure of the multiplicand 

 when multiplying by the second figure of thevmultiplier; drop the last two, when 

 multiplying by the third figure ; and so on. If, however, the tirst partial product 

 has not the desired number, the dropping of figures must be deferred till the 

 third figure of the multiplier is used. 



Examples. Desired the volume V=irr 2 l of a right circular cylinder 

 dimensions are 



in. in. 



r = 6.0428 \ per cent, I = 12.653 ^ per cent. 



The result cannot be more accurate than the least precise factor, which is 

 obviously r. Under the rule, ] per cent computations call for five places of 

 significant figures. Hence we should have to find by multiplication or by five- 

 place logarithms, 



V- 3.1416 x (6.0428)2 x 12.653. 



Note in this connection that the error in V is proportional to twice the 

 percentage error in r, since r enters in the second power, and that in general 

 where a number is raised to any power n the percentage error in the result is 

 increased to n times its value in the data. These rules, however, provide suffi- 

 ciently for such cases. See " Definitions and Explanations." 



Example 4. Desired the volume F = irrH of a right circular cylinder 

 whose dimensions are given as 



r = 6.043 0.017 inches, I 12.653 0.038 inches. 



Under the general principle of retaining places to correspond to two uncer- 

 tain figures in the result in the data, r should have four places and I five places, 

 judging from their stated precision. But the weaker quantity fixes the number 

 of places in the result, so that we should use but four places : 



V= 3.142 x (6.043)2- x 12.65. 



Example 5. Desired the ratio of the radius to the length in each of the 

 Examples 2, 3, and 4. 



The number of places of figures to be used would be respectively four, 

 five, and four, just as in the above solutions. A factor in the denominator fol- 

 lows precisely the same rule as to places of significant figures as the one in the 

 numerator. To contrast the ordinary and shortened solutions, the following are 

 given: 



12.65)6.043(0.4777 



5060 

 "9830 

 8855 



975 

 8855 



8950 

 8855 



9830 



8855 



975 



889 



86 



