414 H. A. Newton— Effect upon the earth's velocity 



greater than 6 and less than d+dd with the earth's motion con- 

 stitute a fraction of the whole group expressed by ^sinddd. 

 Their united action upon the earth at right angles to the earth's 

 motion is zero because of symmetry. Each meteor's total 

 action multiplied by cos <p gives the effective action along the 

 line of the earth's motion. If we represent as before the den- 

 sity of the whole system by d, and the earth's retardation per 

 second in feet per second by p v we have instead of (6) the 

 formula, 



Sdgr^ 2 r n sintfcosffi , 



c+l -, .. , , P ! 1 + 



whence p= g ^ J — -_ log — -^—dx. (8) 



I 



±(c-l) 



K) 



Since cc does not vary through zero the lower limit must be 

 taken arithmetically positive, whatever be the value of c. The 

 value of the definite integral will be different according as c is 



greater than unity, or less than unity. The factor 1+^— 4 



may be considered equal to unity except in the cases wh^re 

 c — 1 and d are both very small and P not very large. The 

 indefinite integral then becomes, 



2(x i — 1+c 2 ) . Pec 2 4(c 2 — 1) 4(/S 2 + l-c 2 ) .x 



~ ~ lo g "I 7* + — " a L tan" 1 -* 



x & x' + ft* x fi (i 



Hence when c<^l 



^=^^]iogp+iog(i-c 2 )-iio g ((i+ c r+n((i--c) 2 +>)+i 



2|?c (5 2 + l— c 2 f v y 



or p I = (J(A;logP — A), 



where & is a constant. (=6*83), and A is a function of c. But 

 if cT>l we get 



Bdgr'Fl, C+l ,, (c-l) 2 + (? 2 1 



A = ^ l°2f h i lOg - -t-5 35 



' x r | & c— 1 J & (c + l) 2 + (5 2 c 



Hence o^dA' where A' is a function of c but independent of 

 P 7 . 



12. The results deducible from (9) and (10) may be thus 

 stated as a theorem. 



