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SOLUTIONS OF A SMITH’S PRIZE PAPER FOR 1871. 
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13. Explain the course of the geodesic lines on a spheroid of revolution: and in 
particular show that the condition is satisfied in virtue of which any geodesic line, con 
sidered as starting from a given point, ceases at some point of its course to he a 
shortest line. 
From each point on the surface a geodesic line may be drawn in any direction 
whatever along the surface, that is, through each point of the surface there is a singly 
infinite series of geodesic lines. A geodesic line undulates (in the manner of a 
sinusoid) between two parallels equidistant from the equator on opposite sides thereof; 
viz. considering it as starting from a point A on the equator, it arrives at a point 
V on the upper parallel (there touching the parallel), and passes downwards to cut 
the equator at A', and thence arrives at a point V' on the lower parallel (there 
touching the parallel), and again passes upwards to meet the equator at A", and so 
on; the arcs AV, VA', A'V', V'A", &c., being similar and equal to each other (differing 
only in position). The equatoreal arc AA' (= A'A" = &c.) or difference of the longitudes 
A, A', is always less than 180°, its value increasing with the inclination at which the 
geodesic line cuts the equator, viz. when this angle is indefinitely small, the arc is 
= -180° (c, a the polar and equatoreal axes respectively), and as the inclination 
ct 
becomes indefinitely near to 90', the value of the arc becomes indefinitely near 180°. 
If the arc in question is commensurable with 180°, the geodesic line will be, it is 
clear, a closed curve; but if not, then it is not a closed curve, but proceeds undulating 
for ever between the two parallels. In the limiting case where the inclination is = 90°, 
the geodesic line is obviously a meridian. 
Considering a geodesic line starting in a given direction from a point A, and 
the geodesic line from the same point A in the consecutive direction, it appears from 
the foregoing account of the configuration of the lines, that the two lines will inter 
sect each other in general an indefinite number of times: supposing that they first 
intersect in a point K, then by a general theorem of Jacobi’s, the geodesic line AK 
is a shortest line from A to any point nearer than K, but it is not a shortest line 
from A to any point beyond K.