content_image/351765.jpg content_image/351773.jpg
## 2. 3rd order flow on S 2 

<img src='content_image/351765.jpg'>

Proposition III.2 Inclusion of the 3 rd order terms for the flow on the canonical two- sphere, with any choice of initial radius ( a ′ 0 ≥ 1 ) and α ′ always gives rise to an ancient solution. 

The solution of the flow equation, taken up to 3 rd order, will be: 

<img src='content_image/351773.jpg'>

where C = a ′ 2 0 ∓ 1 2 ln(4 a ′ 4 0 ± 4 a ′ 2 0 +5) ∓ 3 4 tan − 1 (2 a ′ 2 0 ± 1) 2 .The ± signs refer to α ′ > 0 and α ′ < 0 respectively. 

From eqn.14 and the figures the proposition can be proved without much difficulty. Fig.2 illustrates the conclusions in the proposition. Figures on the left and right are for α ′ > 0 and α ′ < 0 respectively. 

