The President's Address. By E. M. Nelson. 
163 
of two plano-convex lenses having their plane surfaces in contact ; the 
thickness of each must be found and the results added together. If 
the lens is a converging meniscus its thickness will be the difference 
between the thickness of the two pianos. 
Examine 8. — Let the lens be biconvex, the radii being 1*16 and 
2, the diameter d being 2. 
* = 1*16-^1:346-1 = 1*16- V t 3460* = 1*16- *588= *572. 
t' = 2 - */4 ~ 1 = 2 - 1-732 = -268. 
t + t' = *572 + *268 = *840 = the thickness of the lens. 
The third method is to divide the diameter by twice the radius. 
This quotient is a natural sine, which must be looked out in the 
column of natural sines in Chambers’ Tables ; the line in which this 
figure occurs must be followed across, the page until the column of 
versed sines is reached. Take this versed sine and multiply it by the 
radius ; the product will be the thickness of the plano-convex. A 
biconvex or a meniscus must be divided up into pianos as in the 
previous case. 
As the division and multiplication can be performed by slide rule, 
this method is very rapid. 
Example 9. — Find the thickness of a plano-convex lens whose 
diameter d = 2, and radius r = 1*5. 
Here = • 6666 = natural sine of 41° — 48'. The natural 
2 r 
versed sine of this angle is • 255. Then * 255 x 1 * 5 = * 382, the 
thickness. 
When the radius is large in proportion to the diameter, an increase 
in the diameter will only cause a small relative increase in the thick- 
ness, but as the radius approaches the diameter in equality a small 
increase in the diameter will add considerably to the thickness. In 
achromatic combinations the limit of utility is reached when the 
diameter of the combination is equal to 2 r sin 60° = 1 *732 r. 
Example 10. — The radius of the contact curve of the Steinheil 
triple (Example 7) is '413. What is the greatest diameter that 
ought to be given to the combination? Ans. '413 X 1'732 = *715. 
But this is a maximum, in practice it would be better to use 1 * 6 r, 
the diameter in this case would be * 66. 
Having by one of the above plans determined the thicknesses of 
the lenses, we must proceed by the Gauss method to find the focal 
lengths of the combination. The data are the refractive indices /x, /x', 
* Note, great care must be exercised in looking out the square root ; for the 
square root of 346 is 18*60, while that of 3460 is 58*82. Now, as the square root (4’ 
*346 is *5882, it is necessary to look it out in the four figure columns as 3460, and 
then alter the decimal place. If, on the other hand, the figure had been 3 '46, it 
must be looked out as 346, and the decimal place altered to 1*860. 
