Plumb-line in the Hawaiian Islands. 313 



of the integral calculus ceases. But the computation of the 

 resultant attraction of the whole mountain is further facilitated, 

 by so choosing the distances between the circles and lines, that 

 each compartment has a component in the direction of the 

 meridian depending only on its height. Then the attraction 

 of one compartment has only to be multiplied by the sum of 

 the heights of the compartments. These conditions are real- 

 ized by taking the azimuths of the radial lines, such that their 

 sines are in arithmetical progression, and the lines themselves, 

 such that their lengths are in geometrical progression. 

 The formula is. 



J a, *S r , t/ 



(r 2 + 2 2 )^ 



and the final deflection is, 



W'-U^Kl [2(h) -2(H)] 



In calculating the deflection at Kaupo the whole island is 

 considered in three parts, all of which lie to the north. The 

 first is comprised between the prime vertical through the sta- 

 tion and a semi-circumference of 10 miles radius. The second 

 is also limited by the prime vertical and is a semi-circular ring 

 included between radii of 10 and 26 miles. The third takes 

 in that part lying beyond the valley, known as West Maui. 



In the first part the ten radii are in arithmetical progression 

 the common difference being one mile: in the second they 



11 . . /11V 1 



have a ratio of — : the radius of the nth circle being 10 X ( — ) 



10 * \10/ 



(11)* 

 or > ' , miles. In the third they are chosen arbitrarily de- 

 pending on the configuration of the ground. 



Radial lines are now drawn from Kaupo as a center and 

 making angles with the meridian line whose sines have a con- 

 stant difference of one-tenth. This divides the whole island 

 into 318 compartments, the mean heights of which are de- 

 termined from contour lines.* The first series of circles 

 extends to the summit and includes the great crater. 

 Within this limit, which makes up about J the whole surface 

 of the island, we have .more than -f the whole attractive force. 

 The second series goes to the valley, and practically includes 

 all the remainder of the attraction. In computing the effect 

 of West Maui the mountain is divided into two segments and 

 a mean height is assumed for each : giving those parts nearest 

 the attracted point greater weight. These heights are about 



* Kindly furnished by Prof. C. H. Hitchcock. 



